PROBABILITY David Kauchak CS451 – Fall 2013

Admin Midterm Grading Assignment 6 No office hours tomorrow from 10-11am (though I’ll be around most of the rest of the day)

Basic Probability Theory: terminology An experiment has a set of potential outcomes, e.g., throw a dice, “look at” another sentence The sample space of an experiment is the set of all possible outcomes, e.g., {1, 2, 3, 4, 5, 6} For machine learning the sample spaces can very large

Basic Probability Theory: terminology An event is a subset of the sample space Dice rolls  {2}  {3, 6}  even = {2, 4, 6}  odd = {1, 3, 5} Machine learning  A particular feature has a particular values  An example, i.e. a particular setting of features values  label = Chardonnay

Events We’re interested in probabilities of events  p({2})  p(label=survived)  p(label=Chardonnay)  p(parasitic gap)  p(“Pinot” occurred)

Random variables A random variable is a mapping from the sample space to a number (think events) It represents all the possible values of something we want to measure in an experiment For example, random variable, X, could be the number of heads for a coin Really for notational convenience, since the event space can sometimes be irregular spaceHHHHHTHTHHTTTHHTHTTTHTTT X

Random variables We’re interested in probability of the different values of a random variable The definition of probabilities over all of the possible values of a random variable defines a probability distribution spaceHHHHHTHTHHTTTHHTHTTTHTTT X XP(X) 3P(X=3) = 1/8 2P(X=2) = 3/8 1P(X=1) = 3/8 0P(X=0) = 1/8

Probability distribution To be explicit  A probability distribution assigns probability values to all possible values of a random variable  These values must be >= 0 and <= 1  These values must sum to 1 for all possible values of the random variable XP(X) 3P(X=3) = 1/2 2P(X=2) = 1/2 1P(X=1) = 1/2 0P(X=0) = 1/2 XP(X) 3P(X=3) = -1 2P(X=2) = 2 1P(X=1) = 0 0P(X=0) = 0

Unconditional/prior probability Simplest form of probability is  P(X) Prior probability: without any additional information, what is the probability  What is the probability of a heads?  What is the probability of surviving the titanic?  What is the probability of a wine review containing the word “banana”?  What is the probability of a passenger on the titanic being under 21 years old?  …

Joint distribution We can also talk about probability distributions over multiple variables P(X,Y)  probability of X and Y  a distribution over the cross product of possible values MLPassP(MLPass) true0.89 false0.11 EngPassP(EngPass) true0.92 false0.08 MLPass AND EngPassP(MLPass, EngPass) true, true.88 true, false.01 false, true.04 false, false.07

Joint distribution MLPass AND EngPassP(MLPass, EngPass) true, true.88 true, false.01 false, true.04 false, false.07 What is P(ENGPass)? Still a probability distribution  all values between 0 and 1, inclusive  all values sum to 1 All questions/probabilities of the two variables can be calculate from the joint distribution

Joint distribution Still a probability distribution  all values between 0 and 1, inclusive  all values sum to 1 All questions/probabilities of the two variables can be calculate from the joint distribution MLPass AND EngPassP(MLPass, EngPass) true, true.88 true, false.01 false, true.04 false, false.07 How did you figure that out? 0.92

Joint distribution MLPass AND EngPassP(MLPass, EngPass) true, true.88 true, false.01 false, true.04 false, false.07

Conditional probability As we learn more information, we can update our probability distribution P(X|Y) models this (read “probability of X given Y”)  What is the probability of a heads given that both sides of the coin are heads?  What is the probability the document is about Chardonnay, given that it contains the word “Pinot”?  What is the probability of the word “noir” given that the sentence also contains the word “pinot”? Notice that it is still a distribution over the values of X

Conditional probability x y In terms of pior and joint distributions, what is the conditional probability distribution?

Conditional probability Given that y has happened, in what proportion of those events does x also happen x y

Conditional probability Given that y has happened, what proportion of those events does x also happen What is: p(MLPass=true | EngPass=false)? MLPass AND EngPassP(MLPass, EngPass) true, true.88 true, false.01 false, true.04 false, false.07 x y

Conditional probability What is: p(MLPass=true | EngPass=false)? MLPass AND EngPassP(MLPass, EngPass) true, true.88 true, false.01 false, true.04 false, false.07 = Notice this is very different than p(MLPass=true) = 0.89

Both are distributions over X Conditional probability Unconditional/prior probability MLPassP(MLPass) true0.89 false0.11 MLPassP(MLPass|EngPas s=false) true0.125 false0.875

A note about notation When talking about a particular assignment, you should technically write p(X=x), etc. However, when it’s clear, we’ll often shorten it Also, we may also say P(X) or p(x) to generically mean any particular value, i.e. P(X=x) = 0.125

Properties of probabilities P(A or B) = ?

Properties of probabilities P(A or B) = P(A) + P(B) - P(A,B)

Properties of probabilities P(  E) = 1– P(E) More generally:  Given events E = e 1, e 2, …, e n P(E1, E2) ≤ P(E1)

Chain rule (aka product rule) We can view calculating the probability of X AND Y occurring as two steps: 1.Y occurs with some probability P(Y) 2.Then, X occurs, given that Y has occurred or you can just trust the math…

Chain rule

Applications of the chain rule We saw that we could calculate the individual prior probabilities using the joint distribution What if we don’t have the joint distribution, but do have conditional probability information:  P(Y)  P(X|Y) This is called “summing over” or “marginalizing out” a variable

Bayes’ rule (theorem)

Bayes’ rule Allows us to talk about P(Y|X) rather than P(X|Y) Sometimes this can be more intuitive Why?

Bayes’ rule p(disease | symptoms)  For everyone who had those symptoms, how many had the disease?  p(symptoms|disease) For everyone that had the disease, how many had this symptom? p( label| features )  For all examples that had those features, how many had that label?  p(features | label) For all the examples with that label, how many had this feature  p(cause | effect) vs. p(effect | cause)

Gaps I just won’t put these away. These, I just won’t put away. V direct object I just won’t put away. gap filler

Gaps What did you put away? gap The socks that I put away. gap

Gaps Whose socks did you fold and put away? gap Whose socks did you fold ? gap Whose socks did you put away? gap

Parasitic gaps These I’ll put away without folding. gap These without folding. These I’ll put away. gap

Parasitic gaps These I’ll put away without folding. gap 1. Cannot exist by themselves (parasitic) These I’ll put my pants away without folding. gap 2. They’re optional These I’ll put away without folding them. gap

Parasitic gaps ugs-parasitic-gap/

Frequency of parasitic gaps Parasitic gaps occur on average in 1/100,000 sentences Problem: Maggie Louise Gal (aka “ML” Gal) has developed a machine learning approach to identify parasitic gaps. If a sentence has a parasitic gap, it correctly identifies it 95% of the time. If it doesn’t, it will incorrectly say it does with probability Suppose we run it on a sentence and the algorithm says it is a parasitic gap, what is the probability it actually is?

Prob of parasitic gaps Maggie Louise Gal (aka “ML” Gal) has developed a machine learning approach to identify parasitic gaps. If a sentence has a parasitic gap, it correctly identifies it 95% of the time. If it doesn’t, it will incorrectly say it does with probability Suppose we run it on a sentence and the algorithm says it is a parasitic gap, what is the probability it actually is? G = gap T = test positive What question do we want to ask?

Prob of parasitic gaps G = gap T = test positive Maggie Louise Gal (aka “ML” Gal) has developed a machine learning approach to identify parasitic gaps. If a sentence has a parasitic gap, it correctly identifies it 95% of the time. If it doesn’t, it will incorrectly say it does with probability Suppose we run it on a sentence and the algorithm says it is a parasitic gap, what is the probability it actually is?

Prob of parasitic gaps G = gap T = test positive Maggie Louise Gal (aka “ML” Gal) has developed a machine learning approach to identify parasitic gaps. If a sentence has a parasitic gap, it correctly identifies it 95% of the time. If it doesn’t, it will incorrectly say it does with probability Suppose we run it on a sentence and the algorithm says it is a parasitic gap, what is the probability it actually is?

Prob of parasitic gaps G = gap T = test positive Maggie Louise Gal (aka “ML” Gal) has developed a machine learning approach to identify parasitic gaps. If a sentence has a parasitic gap, it correctly identifies it 95% of the time. If it doesn’t, it will incorrectly say it does with probability Suppose we run it on a sentence and the algorithm says it is a parasitic gap, what is the probability it actually is?